The conjecture that cubic roots lie in the enlarged ring class field

Let $u\in S_d$, write $\nu=u^{1/3}$, and let $c=c(u)$ be the product of the distinct rational primes dividing $u$ but not $d$. Let $M_{3c}$ denote the ring class field associated with the order indexed by $3c$. Enlarged ring class field conjecture. For every $u\in S_d$,

$$\nu\in M_{3c}.$$

The claim gives a uniform ring-class-field containment for the cubic roots considered in the paper. The surrounding discussion records numerical evidence for it and notes that, for fundamental units, it reduces to the earlier conjecture concerning $M_3$.